A Kind of Identities Involving Complete Bell Polynomials
2017-03-14
(1.Department of Mathematics,Luoyang Normal College,Luoyang 471934,China;2.Department of Mathematics and Physical,Luoyang Institute of Science and Technology,Luoyang 471023,China)
§1.Introduction
Many interestingq-series identities can be obtained by derivative,see[3-6].Ismail[3]obtained the two following identities by the method of higher order derivatives.
and
In this paper we will establish a formula of higher derivative byFa`a di Brunoformula,and apply it to some exiting results to get some identities involving complete Bell polynomials.Especially we shall give another expression of(1.1)and(1.2)by complete Bell polynomials.
First we give some definitions and lemmas which will be useful throughout this paper.
Theq−shifted factorial is defined by
Whenn→∞,we de fine
The falling factorialzof orderkis defined by
Theq-gamma function Γq(x)is defined by
and
The generalizedq-binomial coefficients are defined by
The generalizedq-harmonic numbers can be defined by
The(exponential)partial Bell polynomialsBn,k=Bn,k(x1,x2,···,xn−k+1)are defined by
Lemma 1[1,p.134TheoremA]The partial Bell polynomials have integral coefficients,are homogeneous of degreek,and of weightn;their exact expression is
where the summation takes place over all integersc1,c2,···,≥0,such that
Lemma 2[1,p.137TheoremA(Fa`adiBrunoformula)]Letfandgbe two formal(Taylor)series:
and lethbe the formal(Taylor)series of the composition ofgbyf.
Hence,coefficientshnare given by the following expression:
where theBn,kare the exponential Bell polynomials.
§2.Main Results and Their Proofs
Theorem 1Letf(x)be analytic function and
Suppose thath(m+1)(x)is themorder derivative ofh(1)(x)form∈N.Then
ProofBy(2.1),we have
So
whereCis an arbitrary constant.By(2.2),we have
On one hand,let
It is easy to see that
On the other hand,byFa`a di Brunoformula,we have
Comparing the coefficients oftn/n!in(2.3)and(2.4),we complete the proof of the theorem.
Corollary 1Leta∈R,n,p∈N and|q|<1.Then
ProofWe have
and
By Theorem 2.1,we complete the proof of the corollary.
Corollary 2Leta∈R,n,p∈N and|q|<1.Then
ProofWe have
and
By Theorem 2.1,we complete the proof of the corollary.
Corollary 3Leta∈R,n,p∈N,|q|<1 andz/=0.Then
where
ProofWe have
and
By Theorem 1,we complete the proof of the corollary.
Corollary 4Leta∈R,n,p∈N,|q|<1 andz/=0.Then
ProofWe have
and
By Theorem 1,we have
By Lemma 1,(2.5)can be written as
The proof of the corollary is completed.
§3.Applications
Theorem 2Letm,n∈N and|q|<1.Then
ProofThe following identity is[3,(2.6)]
it can be rewritten as follows.
Differentiating(3.3)with respect to the variablezformtimes,and by Corollary 0.2 we complete the proof of the theorem.
By(1.1)and(3.1)we get
Corollary 5Letm,n∈N and|q|<1.Then
Theorem 3Letm,n∈N and|q|<1.Then
ProofThe following identity is[3,(2.7)]
it can be rewritten as follows.
Differentiating(3.6)with respect to the variablezform−1 times,and by Corollary 0.2 we complete the proof of the theorem.
Comparing the coefficients ofxj−(−1)jon the right side of(1.2)and(3.4),we obtain
Corollary 6Letm,n∈N and|q|<1.Then
MacMahon[4,vol.2,p.323]get
and(3.7)can be considered as a finite form of the well-known Jacobi triple product identity.
Differentiating this identity with respect to the variablex,then we have
Theorem 4Letm,n,i∈N and|q|<1.Then
whereh(i)=
ProofTakingx→xq−1in(3.7),we have
differentiating(3.7)with respect to the variablexforstimes and by Corollary 1 and Corollary 3,we complete the proof of the theorem.
Takings=1 in Theorem 4,then
Corollary 7Letm,n∈N and|q|<1.Then
Takingx=in Corollary 7,then
Corollary 8Letm<n∈N and|q|<1.Then
Corollary 9Letm∈N and|q|<1.Then
ProofTakingm=nin Corollary 7,we have
Takingin(3.10),we complete the proof of the corollary.
Corollary 10Letm∈N and|q|<1.Then
ProofApplying the operatorto Corollary 7,we have
By the same method as Corollary 9,we have
we complete the proof of the corollary.
General methods of derivation of q-series identities are given in Theorem 0.1 and relevant corollaries.By these methods a kind ofq-series identities withq-Harmonic numbers and complete Bell polynomials are established.As application,we generalized some important q-series identities by higher derivative and got some interesting results in the third part.Whenq→1,a series of combinatorial identities withq-Harmonic numbers and complete Bell polynomials can be found.Some of combinatorial identities play important roles in number theory and analysis.
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杂志排行
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